151 research outputs found
Formation of naked singularities in five-dimensional space-time
We numerically investigate the gravitational collapse of collisionless
particles in spheroidal configurations both in four and five-dimensional (5D)
space-time. We repeat the simulation performed by Shapiro and Teukolsky (1991)
that announced an appearance of a naked singularity, and also find that the
similar results in 5D version. That is, in a collapse of a highly prolate
spindle, the Kretschmann invariant blows up outside the matter and no apparent
horizon forms. We also find that the collapses in 5D proceed rapidly than in
4D, and the critical prolateness for appearance of apparent horizon in 5D is
loosened compared to 4D cases. We also show how collapses differ with spatial
symmetries comparing 5D evolutions in single-axisymmetry, SO(3), and those in
double-axisymmetry, U(1)U(1).Comment: 5 pages, 5 figures, To be published in Phys. Rev.
Asymptotically constrained and real-valued system based on Ashtekar's variables
We present a set of dynamical equations based on Ashtekar's extension of the
Einstein equation. The system forces the space-time to evolve to the manifold
that satisfies the constraint equations or the reality conditions or both as
the attractor against perturbative errors. This is an application of the idea
by Brodbeck, Frittelli, Huebner and Reula who constructed an asymptotically
stable (i.e., constrained) system for the Einstein equation, adding dissipative
forces in the extended space. The obtained systems may be useful for future
numerical studies using Ashtekar's variables.Comment: added comments, 6 pages, RevTeX, to appear in PRD Rapid Com
Constructing hyperbolic systems in the Ashtekar formulation of general relativity
Hyperbolic formulations of the equations of motion are essential technique
for proving the well-posedness of the Cauchy problem of a system, and are also
helpful for implementing stable long time evolution in numerical applications.
We, here, present three kinds of hyperbolic systems in the Ashtekar formulation
of general relativity for Lorentzian vacuum spacetime. We exhibit several (I)
weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric
hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's
original equations form a weakly hyperbolic system. We discuss how gauge
conditions and reality conditions are constrained during each step toward
constructing a symmetric hyperbolic system.Comment: 15 pages, RevTeX, minor changes in Introduction. published as Int. J.
Mod. Phys. D 9 (2000) 1
formalism in Einstein-Gauss-Bonnet gravity
Towards the investigation of the full dynamics in higher-dimensional and/or
stringy gravitational model, we present the basic equations of the
Einstein-Gauss-Bonnet gravity theory. We show -dimensional version of
the ADM decomposition including Gauss-Bonnet terms, which shall be the standard
approach to treat the space-time as a Cauchy problem. Due to the quasi-linear
property of the Gauss-Bonnet gravity, we find that the evolution equations can
be in a treatable form in numerics. We also show the conformally-transformed
constraint equations for constructing an initial data. We discuss how the
constraints can be simplified by tuning the powers of conformal factors. Our
equations can be used both for timelike and spacelike foliations.Comment: 15pages, no figure, to appear in PR
Fate of Kaluza-Klein Bubble
We numerically study classical time evolutions of Kaluza-Klein bubble
space-time which has negative energy after a decay of vacuum. As the zero
energy Witten's bubble space-time, where the bubble expands infinitely, the
subsequent evolutions of Brill and Horowitz's momentarily static initial data
show that the bubble will expand in terms of the area. At first glance, this
result may support Corley and Jacobson's conjecture that the bubble will expand
forever as well as the Witten's bubble. The irregular signatures, however, can
be seen in the behavior of the lapse function in the maximal slicing gauge and
the divergence of the Kretchman invariant. Since there is no appearance of the
apparent horizon, we suspect an appearance of a naked singularity as the final
fate of this space-time.Comment: 13 pages including 10 figures, RevTeX, epsf.sty. CGPG-99/12-8,
RESCEU-6/00 and DAMTP-2000-30. To appear in Phys. Rev.
Constraint propagation in the family of ADM systems
The current important issue in numerical relativity is to determine which
formulation of the Einstein equations provides us with stable and accurate
simulations. Based on our previous work on "asymptotically constrained"
systems, we here present constraint propagation equations and their eigenvalues
for the Arnowitt-Deser-Misner (ADM) evolution equations with additional
constraint terms (adjusted terms) on the right hand side. We conjecture that
the system is robust against violation of constraints if the amplification
factors (eigenvalues of Fourier-component of the constraint propagation
equations) are negative or pure-imaginary. We show such a system can be
obtained by choosing multipliers of adjusted terms. Our discussion covers
Detweiler's proposal (1987) and Frittelli's analysis (1997), and we also
mention the so-called conformal-traceless ADM systems.Comment: 11 pages, RevTeX, 2 eps figure
Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system
Several numerical relativity groups are using a modified ADM formulation for
their simulations, which was developed by Nakamura et al (and widely cited as
Baumgarte-Shapiro-Shibata-Nakamura system). This so-called BSSN formulation is
shown to be more stable than the standard ADM formulation in many cases, and
there have been many attempts to explain why this re-formulation has such an
advantage. We try to explain the background mechanism of the BSSN equations by
using eigenvalue analysis of constraint propagation equations. This analysis
has been applied and has succeeded in explaining other systems in our series of
works. We derive the full set of the constraint propagation equations, and
study it in the flat background space-time. We carefully examine how the
replacements and adjustments in the equations change the propagation structure
of the constraints, i.e. whether violation of constraints (if it exists) will
decay or propagate away. We conclude that the better stability of the BSSN
system is obtained by their adjustments in the equations, and that the
combination of the adjustments is in a good balance, i.e. a lack of their
adjustments might fail to obtain the present stability. We further propose
other adjustments to the equations, which may offer more stable features than
the current BSSN equations.Comment: 10 pages, RevTeX4, added related discussion to gr-qc/0209106, the
version to appear in Phys. Rev.
Quasi-spherical approximation for rotating black holes
We numerically implement a quasi-spherical approximation scheme for computing
gravitational waveforms for coalescing black holes, testing it against angular
momentum by applying it to Kerr black holes. As error measures, we take the
conformal strain and specific energy due to spurious gravitational radiation.
The strain is found to be monotonic rather than wavelike. The specific energy
is found to be at least an order of magnitude smaller than the 1% level
expected from typical black-hole collisions, for angular momentum up to at
least 70% of the maximum, for an initial surface as close as .Comment: revised version, 8 pages, RevTeX, 8 figures, epsf.sty, psfrag.sty,
graphicx.st
Adjusted ADM systems and their expected stability properties: constraint propagation analysis in Schwarzschild spacetime
In order to find a way to have a better formulation for numerical evolution
of the Einstein equations, we study the propagation equations of the
constraints based on the Arnowitt-Deser-Misner formulation. By adjusting
constraint terms in the evolution equations, we try to construct an
"asymptotically constrained system" which is expected to be robust against
violation of the constraints, and to enable a long-term stable and accurate
numerical simulation. We first provide useful expressions for analyzing
constraint propagation in a general spacetime, then apply it to Schwarzschild
spacetime. We search when and where the negative real or non-zero imaginary
eigenvalues of the homogenized constraint propagation matrix appear, and how
they depend on the choice of coordinate system and adjustments. Our analysis
includes the proposal of Detweiler (1987), which is still the best one
according to our conjecture but has a growing mode of error near the horizon.
Some examples are snapshots of a maximally sliced Schwarzschild black hole. The
predictions here may help the community to make further improvements.Comment: 23 pages, RevTeX4, many figures. Revised version. Added subtitle,
reduced figures, rephrased introduction, and a native checked. :-
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